Question

Consider a civilization broadcasting a signal with a power of 1.2×104 watts. The Arecibo radio telescope, which is about 300 meters in diameter, could detect this signal if it is coming from as far away as 110 light-years. Suppose instead that the signal is being broadcast from the other side of the Milky Way Galaxy, about 70000 light-years away.

Answer 1

The same power (1.9×10^4 watts) gets diminished by Inverse square law
(ratio of distances)²
= (138/70000)²
= 3.886506 X 10^-6 .
This is the diminution factor for the given sensitivity of the telescope. But at 3.8865 millionth below the sensitivity its is far below detection; it needs to be enhanced by collecting the energy over an area so many times more (by an aperture multiplied by same factor)
1/[3.886506 X 10^-6] = 257298.88
In other words the diameter should have the ratio of square root of this
(70000/138).
This multiplied by 300m dish gives
300 X (70000/138) = 152173.913 m = 152.1739 km.
This should be the aperture of the new telescope for detecting the signal.